46 HlSTOKY OP THE THEORY OF NUMBERS. [CHAP. I (3D ««{ (M) «; (13) 3- (33) 3>.54M9\|™| (32) 3".5.13{Łg (12) S (30) 33.5 (55) (56) 3 (58) P Euler's final list of 61 pairs did not include the pairs a, ft 7, although he had obtained a four times in the body of his paper, viz., in (2), (3-0, (53); 0 twice in (30; 7 in (2). Moreover, these three unlisted pairs occur as VIII, IX, and XIII among the 30 pairs in Euler's362 earlier list, a fact noted on p. XXVI and p. LVIII of the Preface by P. H. Fuss and N. Fuss to Euler's Comm. Arith. Coll., who failed to observe that these three pairs occur in the text of Euler's present paper. Nor did these editors note that the fourth mentioned case of divergence between the two lists is due merely to the misprint3646 of 57 for 47 in (43) of the present list, so that the correctly printed pair XXVIII of the list of 30 is really this (43) and not a new pair, as supposed by them. From the fact that Euler obtained in his posthumous tract97 on amicable numbers the pairs a, ]3 (once on p. 631 and again on p. 633 and finally on p. 635), the editors inferred, p. LXXXI of the Preface, that the tract differs in analysis from the long paper just discussed. But no new pairs are found, while the cases treated on pp. 631-2 are merely problems 1 and 2 of Euler's preceding paper. It is different with p. 634, where Euler started with two numbers like 71 and 5-11 which, by his table, have the same sum, 72, of divisors, and required a number a relatively prime to them such that 71a and 55a are amicable. The single condition is 72ja = (71+55)a, whence Ja:a=7:4. Thus a has the factor 4. If a = 46, where 6 is odd, then fb = b = 1, and the pair 284, 220 results. The case a = 86 is impossible. This method was used in a special way by Kraft363 who limited the numbers from which one starts to a prime and a product of two primes. In the Encyclopedic Sc. Math., I, 3i, p. 59, note 320, it is stated that this posthumous tract contains four pairs not in Euler's list of 61, two pairs being those of Fermat354 and Descartes.355 But these were listed as (2) and (3) by Euler and were obtained by him in case (1L) and attributed to Descartes. E. Waring365 noted that 2nx, 2nyz are amicable if 22n where x, y, z are primes and y — 2n-{-l divides 22n. He cited the first two such pairs of amicable numbers. **eG. Eneatrdm, Bibliotheca Math., (3), 9, 1909, 263. 3MMeditationes algebraicae, 1770, 201; ed. 3, 1782, 342-3.